# Properties

 Label 10T18 Degree $10$ Order $200$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_5^2 : C_8$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(10, 18);

## Group action invariants

 Degree $n$: $10$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $18$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_5^2 : C_8$ CHM label: $[5^{2}:4]2_{2}$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,7,9,3)(2,4,8,6), (1,4,3,2,9,6,7,8)(5,10), (2,4,6,8,10) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$8$:  $C_8$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 5: None

## Low degree siblings

10T18 x 2, 20T56 x 3, 25T20, 40T171 x 3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 4, 1, 1$ $25$ $4$ $( 3, 5, 9, 7)( 4, 6,10, 8)$ $4, 4, 1, 1$ $25$ $4$ $( 3, 7, 9, 5)( 4, 8,10, 6)$ $2, 2, 2, 2, 1, 1$ $25$ $2$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)$ $5, 1, 1, 1, 1, 1$ $8$ $5$ $( 2, 4, 6, 8,10)$ $8, 2$ $25$ $8$ $( 1, 2)( 3, 4, 7, 8, 9,10, 5, 6)$ $8, 2$ $25$ $8$ $( 1, 2)( 3, 6, 5,10, 9, 8, 7, 4)$ $8, 2$ $25$ $8$ $( 1, 2)( 3, 8, 5, 4, 9, 6, 7,10)$ $8, 2$ $25$ $8$ $( 1, 2)( 3,10, 7, 6, 9, 4, 5, 8)$ $5, 5$ $8$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$ $5, 5$ $8$ $5$ $( 1, 3, 5, 7, 9)( 2, 8, 4,10, 6)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $200=2^{3} \cdot 5^{2}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 200.40 magma: IdentifyGroup(G);
 Character table:  2 3 3 3 3 . 3 3 3 3 . . 5 2 . . . 2 . . . . 2 2 1a 4a 4b 2a 5a 8a 8b 8c 8d 5b 5c 2P 1a 2a 2a 1a 5a 4b 4a 4a 4b 5b 5c 3P 1a 4b 4a 2a 5a 8c 8d 8a 8b 5b 5c 5P 1a 4a 4b 2a 1a 8d 8c 8b 8a 1a 1a 7P 1a 4b 4a 2a 5a 8b 8a 8d 8c 5b 5c X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 1 -1 -1 -1 -1 1 1 X.3 1 -1 -1 1 1 A -A -A A 1 1 X.4 1 -1 -1 1 1 -A A A -A 1 1 X.5 1 A -A -1 1 B /B -/B -B 1 1 X.6 1 A -A -1 1 -B -/B /B B 1 1 X.7 1 -A A -1 1 -/B -B B /B 1 1 X.8 1 -A A -1 1 /B B -B -/B 1 1 X.9 8 . . . 3 . . . . -2 -2 X.10 8 . . . -2 . . . . -2 3 X.11 8 . . . -2 . . . . 3 -2 A = -E(4) = -Sqrt(-1) = -i B = -E(8) 

magma: CharacterTable(G);